If \(S\)  is given a complex structure \(J\) , then to any holomorphic quadratic differential \(q=q(z)dz^{2}\) , one may consider the foliation obtained by integrating the line field \(q(v,v) >0.\)  When further given the transverse measure defined by \(\int _{\alpha }| \operatorname{Im}(\sqrt{q})|\) , the resulting measured foliation is called the horizontal foliation of \(q\) . Likewise integrating the line field \(q(v,v) <0\)  and taking the measure \(\int _{\alpha }| \operatorname{Re}(\sqrt{q})|\)  gives the vertical foliation of \(q\) . The theorem of Hubbard-Masur [1]} states that for a fixed Riemann surface \((S,J)\)  and any measured foliation \(\mathcal {F}\)  on \(S\) , there is a unique holomorphic quadratic differential \(q\) , whose horizontal foliation is Whitehead equivalent (i.e. it differs at most by isotopies or expanding or collapsing pronged singularities along straight arcs) to \(\mathcal {F}\) .
